To solve the provided differential equation, apply separation of variables and integrate both sides, using partial fractions as necessary. After integrating, include the constant of integration c to obtain the final solution for y in terms of x.
To solve the differential equation dy/dx = (x⁸ - 2y)/x, one can use the method of separation of variables. We start by multiplying both sides by dx and then dividing both sides by (x⁸ - 2y) to get dy/(x⁸ - 2y) = dx/x. Splitting the fraction on the left side into partial fractions is the next step to proceed with separating the variables completely.
Once the variables are separated, both sides can be integrated to find the general solution of the differential equation. The solution will include an integration constant denoted as c. After integrating, we express y in terms of x and c to get the answer.
The exact steps of integration and partial fraction decomposition involve a higher level of algebra and calculus, which are beyond the 150 words explanation. Nonetheless, the student should proceed by integrating both sides after the variables are separated, apply the appropriate integration techniques, and include the constant of integration c as the final step.